Chapter 7: Mechanisms in Programs and Nature

Section 6: The Phenomenon of Continuity

History [of Central Limit Theorem]

That averages of random numbers follow bell-shaped distributions was known in the late 1600s. The formula for the Gaussian distribution was derived by Abraham de Moivre around 1733 in connection with theoretical studies of gambling. In the late 1700s Pierre-Simon Laplace did this again to predict the distribution of comet orbits, and showed that the same results would be obtained for other underlying distributions. Carl Friedrich Gauss made connections to the distribution of observational errors, and the relevance of the Gaussian distribution to biological and social systems was noted. Progressively more general proofs of the Central Limit Theorem were given from the early 1800s to the 1930s. Many natural systems were found to exhibit Gaussian distributions—a typical example being height distributions for humans. (Weight distributions are however closer to lognormal; compare page 1003.) And when statistical methods such as analysis of variance became established in the early 1900s it became increasingly common to assume underlying Gaussian distributions. (Gaussian distributions were also found in statistical mechanics in the late 1800s.)

From Stephen Wolfram: A New Kind of Science [citation]